Ok, I might be able to point you to a couple of recent review articles or some related matter ... or give my own impression of what is used . A point of clarification though : are you focusing on ODEs and "simplistic" applications or "real-life" research problems, meaning a bit more complex PDEs and so forth ?

one of the most popular is called ODE45. This is a package that is used in MATLAB and SIMULINK. Try looking up ODE45 on google. I believe it uses a smart version of the runge kutta method. By smart I mean that it knows how the adjust the step size so that it computes a solution quick but remains within a certain degree of accuracy. Many of the relative and absolute error tolerance can be user defined and it is a very reliable and accurate solver. All the user has to do is write a function that contains the DE's in state variable form and run the ODE45 command with the specified initial conditions and the time interval of the simulation

My comments are specifically aimed at solving huge systems of (partial) differential equations which arise in modeling of physical systems. I doubt this applies if you're just trying to solve a single equation.

I think there is a trend towards using boundary element methods over finite difference or finite element methods. This means you turn your differential equation into an integral equation and solve that instead. For a long time people liked FEM and FDM because they generated sparse matrices which are easy to solve, as opposed to BEM which ALWAYS generate dense matrices. But now there are fast-solvers and better computers which can handle dense matrices much easier, so BEM are becoming more practical.

Advantages of BEM are that you can solve exterior problems 'exactly', whereas the other methods require some sort of truncation of your mesh. Also, BEM often converge spectrally (exponentially), which is much faster than the average FE method. The downside to them is that it's sometimes difficult/impossible to turn the differential equation into an integral equation.

Yeah, I'm inclined to agree what comes to BEM ... although have to say that when bumping into problems involving nonlinearities FEM is still a preferable choice and BEM faces problems. Although there are ways to tackle these issues. One added group of methods I'm seeing on the rise are meshfree methods, such as element free Galerkin, which have enabled analyses of problems overly difficult even for adaptive FEM & BEM.